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Synopsis:

Multiplication of matrices corresponds to composition of maps, and when f and g are maps so that the target Q of g equals the source P of f, the product f*g is defined, its source is the source of g, and its target is the target of f. The degree of f*g is the sum of the degrees of f and of g. The product is also defined when P != Q, provided only that P and Q are free modules of the same rank. If the degrees of P differ from the corresponding degrees of Q by the same degree d, then the degree of f*g is adjusted by d so it will have a good chance to be homogeneous, and the target and source of f*g are as before.

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